This record has been superseded. The current, updated version is A Tensor-Network Kinetic Solver for Fusion Transport: Matrix-Product-State Compression of the Distribution Function with Exponential Error Decay (DOI 10.5281/zenodo.22645768). Please read and cite the current version. This page stays online so existing citations keep resolving. The kinetic distribution function is highly compressible in a low-rank tensor-network representation, and that yields a practical classical solver, not a storage trick. On a 1D1V BGK test, a matrix-product-state truncation reaches relative-L² error 2×10−⁴ at rank 8 using ~0.19× the dense storage, with the error falling exponentially in rank. Unlike the fault-tolerant-horizon quantum route, this runs today.
This record has been superseded. The current, updated version is A Tensor-Network Kinetic Solver for Fusion Transport: Matrix-Product-State Compression of the Distribution Function with Exponential Error Decay (DOI 10.5281/zenodo.22645768). Please read and cite the current version. This page stays online so existing citations keep resolving. The kinetic distribution function is highly compressible in a low-rank tensor-network representation, and that yields a practical classical solver, not a storage trick. On a 1D1V BGK test, a matrix-product-state truncation reaches relative-L² error 2×10−⁴ at rank 8 using ~0.19× the dense storage, with the error falling exponentially in rank. Unlike the fault-tolerant-horizon quantum route, this runs today.
Kinetic descriptions of fusion plasmas are throttled by the curse of dimensionality: a distribution function on a six-dimensional phase space is intractable to store on a dense grid, let alone to advance in time. We show that the kinetic distribution function is strongly compressible in a low-rank tensor-network (matrix-product-state, or tensor-train) representation and that this yields a practical solver on present-day classical hardware, not merely a storage economy. We cast the problem formally: the discretized distribution is a high-order tensor, its matrix-product-state factorization is exact up to a truncation error bounded by the discarded Schmidt weight (Eckart–Young), and the number of retained modes needed for a target accuracy scales with the entanglement entropy of the state. On a one-dimensional-in-space, one-dimensional-in-velocity Bhatnagar–Gross–Krook (BGK) kinetic test problem, discretized on a periodic N_x=128× N_v=64 grid and advanced by operator-split upwind streaming and implicit relaxation, a matrix-product-state truncation reaches a relative-L^2 error of 2.9×10^-6 at bond dimension =8 while storing 0.19× the dense grid, and the error falls exponentially in the bond dimension — from 1.8×10^-1 at =1 to 2.2×10^-10 at =16 and to the double-precision floor (~10^-14) by =24. The velocity moments inherit the same convergence (density error 1.1×10^-6, temperature error 2.8×10^-6 at =8), and the converged distribution carries a Schmidt entanglement entropy below 0.1 nats — the area-law signature that makes it compressible. Every headline quantity is a frozen anchor in the Kronos de-risking register (gate KX-L1-A16). Beyond the anchor we give the general-dimensional machinery that carries the method to the five- and six-dimensional gyrokinetic problem — matrix-product-operator representations of the streaming and collision maps, dimension-linear tensor-train construction by cross interpolation, and a rank-adaptive, conservative dynamical-low-rank integrator that evolves the state on the low-rank manifold without inverting small singular values — together with the derived exponential-to-linear crossover in both storage and per-step arithmetic that motivates it. The result is a classical, deployable-today complement to the fault-tolerant-horizon quantum kinetic route, and we place it as a cross-cutting, verification-first module inside the Kronos autonomous control architecture: an advisory/supervisory reduced-kinetics solver, exercised twin-first beneath a deterministic safety clamp, that hands authority back to the deterministic floor on out-of-distribution inputs. The single honest gate is that the demonstration is a collision-dominated reduced problem, and the rank growth of the fully nonlinear, collisionless six-dimensional gyrokinetic distribution at reactor parameters remains the open question we scope explicitly.Key results (frozen anchors): v = 2 v; t = 3.27 ×10^-3; N_v = 128 ×64; sigma_1 = 9.4 ×10^-2; sigma_1 = 2.5 ×10^-3; sigma_1 = 4.7 ×10^-5.Methods & codes: CGYRO, TGLF, GENE, Sauter, VQE, ADAPT-VQE.Live verification: 1 gate validator(s) with live-recompute cards (1 reproduced). See the Verification section and data/verification.csv.Related: Paper page · De-risking register · 3D model · Learn more about KronosPart of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689).All numerical values are frozen design-point anchors; see the register.Public research artifact. No proprietary, financial, or supply-chain information is included.Note (REPLACE): This record replaces and supersedes DOI 10.5281/zenodo.22132192; please cite this version.
Resolving unsteady transport phenomena in geometrically complex domains is traditionally constrained by polynomial scaling of computational cost with spatial resolution. While methods based on tensor-network data representations or matrix-product states (MPS) data encodings have emerged as a technique to systematically reduce degrees of freedom, existing formulations do not extend to complex geometries and complex flow physics. Both capabilities are offered by lattice Boltzmann methods, for which we develop a generalized MPS formulation. This development marks a paradigm shift from classical methods that rely on explicit grid refinement for data reduction. Instead, our approach exploits non-local correlations in the MPS representation to systemically compress the global fluid state directly without modifying the underlying grid. We benchmark the proposed solver against classical LBM using three-dimensional flows through structured media and vascular geometries. The results confirm that the MPS formulation reproduces the reference solution with high fidelity while achieving compression ratios exceeding two orders of magnitude, positioning tensor networks or MPS encodings as a scalable paradigm for continuum mechanics on high-performance GPU hardware.
Kinetic descriptions of fusion plasmas are throttled by the curse of dimensionality: a distribution function on a six-dimensional phase space is intractable to store on a dense grid, let alone to advance in time. We show that the kinetic distribution function is strongly compressible in a low-rank tensor-network (matrix-product-state, or tensor-train) representation and that this yields a practical solver on present-day classical hardware, not merely a storage economy. We cast the problem formally: the discretized distribution is a high-order tensor, its matrix-product-state factorization is exact up to a truncation error bounded by the discarded Schmidt weight (Eckart–Young), and the number of retained modes needed for a target accuracy scales with the entanglement entropy of the state. On a one-dimensional-in-space, one-dimensional-in-velocity Bhatnagar–Gross–Krook (BGK) kinetic test problem, discretized on a periodic N_x=128× N_v=64 grid and advanced by operator-split upwind streaming and implicit relaxation, a matrix-product-state truncation reaches a relative-L^2 error of 2.9×10^-6 at bond dimension =8 while storing 0.19× the dense grid, and the error falls exponentially in the bond dimension — from 1.8×10^-1 at =1 to 2.2×10^-10 at =16 and to the double-precision floor (~10^-14) by =24. The velocity moments inherit the same convergence (density error 1.1×10^-6, temperature error 2.8×10^-6 at =8), and the converged distribution carries a Schmidt entanglement entropy below 0.1 nats — the area-law signature that makes it compressible. Every headline quantity is a frozen anchor in the Kronos de-risking register (gate KX-L1-A16). Beyond the anchor we give the general-dimensional machinery that carries the method to the five- and six-dimensional gyrokinetic problem — matrix-product-operator representations of the streaming and collision maps, dimension-linear tensor-train construction by cross interpolation, and a rank-adaptive, conservative dynamical-low-rank integrator that evolves the state on the low-rank manifold without inverting small singular values — together with the derived exponential-to-linear crossover in both storage and per-step arithmetic that motivates it. The result is a classical, deployable-today complement to the fault-tolerant-horizon quantum kinetic route, and we place it as a cross-cutting, verification-first module inside the Kronos autonomous control architecture: an advisory/supervisory reduced-kinetics solver, exercised twin-first beneath a deterministic safety clamp, that hands authority back to the deterministic floor on out-of-distribution inputs. The single honest gate is that the demonstration is a collision-dominated reduced problem, and the rank growth of the fully nonlinear, collisionless six-dimensional gyrokinetic distribution at reactor parameters remains the open question we scope explicitly. Key results (frozen anchors): v = 2 v; t = 3.27 ×10^-3; sigma_1 = 9.4 ×10^-2; sigma_1 = 2.5 ×10^-3; sigma_1 = 4.7 ×10^-5. Methods & codes: CGYRO, TGLF, GENE, Sauter, VQE, ADAPT-VQE. Live verification: 1 gate validator(s) with live-recompute cards (1 reproduced). See the Verification section and data/verification.csv. Related: Paper page · De-risking register · 3D model · Learn more about Kronos Part of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689). All numerical values are frozen design-point anchors; see the register. Public research artifact. No proprietary, financial, or supply-chain information is included. Previous version: DOI 10.5281/zenodo.22132192. Please cite this version.
Tensor networks (TNs), originally developed for simulating many-body quantum systems, provide a systematic framework for approximating high-dimensional fields. This is achieved by factorizing the field into interconnected tensors with small bond dimensions, thereby restricting the correlations captured across field bipartitions. Belonging to the family of TNs, the matrix product state (MPS) ansatz is utilized here as a reduced-order modeling framework to construct truncated representations of isotropic turbulent flow data. Two direct numerical simulation (DNS) datasets are considered: the hydrodynamic field of an incompressible three-dimensional flow, and a conserved Fickian scalar in a similar flow. Each field is encoded as an MPS through a sequence of singular value decompositions (SVDs) in which small singular values are discarded. The truncated representation is contracted back to the full grid, and the resulting reconstructed field is compared against DNS. An interleaved ordering of the spatial tensor indices of the transport variables is applied prior to decomposition in order to localize the dominant inter-tensor correlations. Velocity reconstructions achieve 99.8% fidelity using only 5% of the original DNS memory, while the scalar field reaches the same fidelity at 15% memory usage. A wide range of lower- and higher-order statistics, including velocity gradients, dissipation, and structure functions, are systematically examined. At these compression levels, the total kinetic energy and the scalar energy are both recovered within 0.2% relative error, while the mean dissipation and mean scalar dissipation remain within approximately 10% of the DNS generated values. These findings support the suitability of MPS for scalable reduced-order analysis of complex turbulent datasets and motivate further exploration of TN-based methods in computational turbulence.
In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor (which we will see are all the same object in this context) has \emph{low rank} may be ill-justified. There are many natural instances where the object in question has high rank with respect to the classical notions of rank: matrix rank, tensor rank, multilinear rank --- the latter two being the most straightforward generalizations of the former. To remedy this, we show that one may vastly expand these classical notions of ranks: Given any undirected graph G, there is a notion of G-rank associated with G, which provides us with as many different kinds of ranks as there are undirected graphs. In particular, the popular tensor network states in physics (e.g., \textsc{mps}, \textsc{ttns}, \textsc{peps}) may be regarded as functions of a specific G-rank for various choices of G. Among other things, we will see that a function, matrix, or tensor may have very high matrix, tensor, or multilinear rank and yet very low G-rank for some G. In fact the difference is in the orders of magnitudes and the gaps between G-ranks and these classical ranks are arbitrarily large for some important objects in computer science, mathematics, and physics. Furthermore, we show that there is a G such that almost every tensor has G-rank exponentially lower than its rank or the dimension of its ambient space.
Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high-resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as a MPS within some tolerable error threshold. In this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations for systems with one coordinate in space and one coordinate in velocity, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher-dimensional problems.